I'm trying to setup a very minimal BBB. I have the ATMega328 in the socket on the board (both in the correct orientation), a voltage regulator (non-LDO) with the appropriate capacitors, power lines ran to where the power jack would go, the three resistors near the LED, the LED itself, and a 16MHz crystal with appropriate capacitors.

I tried shorting the diode spot near the power jack, but that didn't work (the shorting jumper got hot). So I tried leaving it open... nothing. I've already uploaded the "Blink" demo sketch, since I pulled this ATMega328 chip from a Duemilanove board (where it worked). I'm using a 9V power supply... not sure where I've gone wrong.

Sorry about the slow reply. Some email must have fallen through the cracks.
I'm not quite sure what you're trying to do. Are you actually using a BBB board, or just building a breadboard Arduino / BBB?

Your setup sounds about right. Do you have the shunt in that says "USB/EXT" - it won't run without that - or shorting the holes somehow

The diodes are configured reverse polarity across the power line - so shorting them will create a direct short - not surprised things got hot.

Pull out your meter and check for some voltages at proper places - see the schematic.